We are organizing the 8th Complex Systems REU (CSREU) this coming Summer 2020. This REU is a six week program designed to engage students in research problems in dynamical systems and pattern formation, motivated by applications to the physical sciences. We are seeking four students to work on two distinct projects in close collaboration with faculty mentor Arnd Scheel as well as graduate student mentors Montie Avery, Olivia Cannon, and Sally Jankovic. Students will share an office in the School of Mathematics and have daily access to both faculty and graduate student mentors.
Prerequisites. Potentially relevant fields of math include dynamical systems, bifurcation theory, PDE theory, numerical computation, and harmonic analysis. The focus of the projects will be shaped by student interest, and prior background in these fields is not required.
Introductory coursework in dynamical systems, ordinary or partial differential equations is required. Higher level coursework and/or some knowledge of numerical computation is helpful for some aspects of the projects, but not required. No prior research experience is required.
Pattern formation and complex systems. Pattern formation is the study of general mechanisms leading to the appearance of simple or complex spatial patterns. It is motivated by the existence of similar patterns in seemingly dissimilar systems (e.g. animal coat markings, vegetation patterns, phase separation problems, convection patterns).
Participants will receive a stipend of $3,000 as well as up to $2,000 for travel and living expenses. Students will need to be US citizens or permanent residents in order to receive funding. Women and underrepresented minorities are especially encouraged to apply.
I regularly offer research opportunities for undergraduate students. Please contact me if you are interested in doing research in the general area of dynamical systems, pattern formation, and nonlinear waves. Some students in past programs have been supported from other sources such as Undergraduate Research Scholarships at the University of Minnesota.
If you are interested in undergraduate research outside the scope of this NSF-sponsored program or do not qualify as an applicant, please contact me via
and be prepared to send
A list of upper level math courses you have completed or
are taking, along with your grades in these courses
A statement of your interest including your availability.
I will then arrange for a meeting to discuss frameworks and possible projects.
What to expect.
The setup will be similar to previous REU programsthat took place in the summers 2009, 2011, 2012, 2014, 2015, 2017, and 2018; see the advertisement for past experiences 2009, 2011, 2012, 2014, 2017, and 2018.
What do students do after the REU?
Previous students have subsequently gone on to graduate school at Brown, CalTech, Columbia, Cornell (2), Harvard, Maryland, McGill, Moscow, NYU, Oxford, Princeton, Purdue U, U Chicago, UC Boulder, UC Irvine, UIUC, U Minnesota (2); three received NSF Graduate Fellowships.
What happened in the last two projects?
Growing Stripes, With and Without Wrinkles
This project fit into a longstanding effort to understand pattern formation in growing domains. We studied a partial differential equation model in which pattern formation is driven at an interface - to the left of the interface, patterns can form, but to the right, they cannot. The central question was: how does the speed of the interface determine the features of the pattern left behind in its wake? We were specifically interested in answering this question in the presence of a mechanism known as the zigzag instability, which itself selects angles of stripe patterns. We formally reduced this problem to studying an equation for the angle of the pattern itself, which ultimately becomes an ordinary differential equation, allowing for some explicit analysis. We found that oblique stripe patterns, at an angle to the interface, exist for small speeds. From this starting point, we continued these oblique stripes numerically, and found that they terminate in a bifurcation which gives way to periodic wrinkling of stripe patterns. We finished the project in collaboration with Ryan Goh (an alum of this REU, now assistant professor at Boston University!) and computed a 3D bifurcation diagram which systematically encodes the structure of these stripe patterns depending on the interface speed. See figure on the left for emerging patterns.
Pinning and depinning: from periodic to chaotic and random media
In this project, we studied propagation of waves on discrete lattices, modeling for instance electrical impulses in nerve fibers. Our work focused on the phenomenon of pinning, in which for certain values of system parameters, waves are unable to propagate, becoming stuck or "pinned". As a parameter is increased past some critical value, the wave is suddenly able to propagate, and so undergoes depinning. The central goal in our project was to understand how the speed of the wave depends on system parameters near this depinning threshold, particularly in complex media. We modeled this complexity by considering media which are generated by discrete dynamical systems. In this way, we built a framework that captures transitions from constant media to periodic media and then to chaotic, highly complex media. This framework also presented an abundance of concrete examples for numerical study. We demonstrated that depinning asymptotics are universal: the dependence of the wave speed on the parameter near the depinning threshold is determined only by a measure of the underlying dimension of the medium. See figure on the right for depinning thresholds in a cat-map medium.
Publications. Results of previous programs lead to the following journal publications.
M. Avery, R. Goh, O. Goodloe, A. Milewski, A. Scheel